Should I be concerned about this pivot point?
The short answer is good news: This doesn't matter, because all of this kind of stuff won't really affect UV unwrapping or texturing, and you can go back and change it later if you need to. You might start to run into problems if you have a non-uniform scale, so it's best to apply a scale before UV-unwrapping. Blender will give you a warning if this is the case.

When you apply location, that means you are setting the origin of that object to the origin of the x-y-z volume, so that is exactly expected behavior.

Where you want that pivot point mostly depends on how you intend to animate your mesh. If you're using an armature, and mapping parts of your mesh to bones, then it doesn't really matter where the object origins are. If you want to be able to animate that hat as its own object, it probably makes more sense to keep the origin in the center of the geometry, because that's the point about which you'll animate rotations.

You can change your origin in the Object menu under Set Origin. Typically you'll either Set Origin to Geometry, or Set Origin to 3D cursor if you need the origin to be somewhere specific, like at the hinge of a door as opposed to its center.

So for example, if you wanted to keep the hat where it is, with the hat's origin at its center of geometry, here is how you would go about it.


Select the hat. Object > Set Origin > To Geometry
SHIFT+click the character to select it as well and hit CTRL+P > Set Parent.